Variants of the Mattila Integral, Measures with Nonnegative Fourier Transforms, and the Distance Set Problem

Abstract

Suppose mu is an element of M(R-N) is a measure with parallel to mu parallel to > 0, sigma is surface measure on the unit sphere Sn-1 subset of R-n, and phi is an element of L-2 (Sn-1) is a function parallel to phi parallel to(L2) ((sn-1)) > 0. If parallel to(mu) over tilde parallel to L-2(R-N) < infinity, then supp mu it has positive Lebesgue measure. We ask the question, what can we say about supp mu under the weaker assumption_x000D_ integral(infinity)(0)vertical bar integral(sn-1) (mu) over bar (r theta)phi(theta)d sigma(theta)vertical bar(2) r(n-1) dr < infinity ?_x000D_ We give an answer in the case phi is an element of C-infinity(Sn-1) and relate our result to Falconer's distance set problem. Our line of investigation naturally leads us to the study of measures with nonnegative Fourier transforms, which we also relate to Falconer's distance set problem in both the Euclidean setting and in vector spaces over finite fields. As an application of our results, we give a new proof of the Erdiis-Volkmann ring conjecture.